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Chapter 5: Problem 48

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with \(y\) -coordinates given. $$f(x)=x^{3}+3 x-4, y=0,1,2$$

Short Answer

Expert verified
The inverse points are approximately (0,1.35), (1,1.41), and (2,1.51).

Step by step solution

01

Graph the Function

First, graph the function \( f(x) = x^3 + 3x - 4 \) using a graphing calculator. This image will help us visualize the behavior of the function. Note how the curve progresses through various regions on the Cartesian plane.
02

Understand the Inverse Relationship

Remember that if \((a, b)\) is a point on the graph \( y = f(x) \) of a function, then \((b, a)\) is a point on the graph of its inverse. We will find specific points on the inverse graph given particular \( y \)-coordinates.
03

Identify Points on the Graph

For each given \( y \)-coordinate (0, 1, and 2), identify the corresponding \( x \)-coordinates by solving \( f(x) = y \). These \( x \) values form the \( y \)-coordinates of points on the graph of the inverse.
04

Find Points for \( y = 0 \)

Set the equation \( f(x) = 0 \), which simplifies to \( x^3 + 3x - 4 = 0 \). Solve for \( x \). The solution will give us a point \((0, x_1)\) for the inverse graph.
05

Find Points for \( y = 1 \)

Set the equation \( f(x) = 1 \), which simplifies to \( x^3 + 3x - 5 = 0 \). Solve for \( x \). The solution will give us a point \((1, x_2)\) for the inverse graph.
06

Find Points for \( y = 2 \)

Set the equation \( f(x) = 2 \), which simplifies to \( x^3 + 3x - 6 = 0 \). Solve for \( x \). The solution will give us a point \((2, x_3)\) for the inverse graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Calculators
Graphing calculators are powerful tools to visualize mathematical functions quickly. When tackling problems like graphing functions and their inverses, a graphing calculator becomes essential. Its ability to plot a graph electronically allows you to see the function's behavior without manually plotting each point.
Using a graphing calculator involves:
  • Inputting the function: Simply enter the function such as \(f(x) = x^3 + 3x - 4\) into the calculator.
  • Viewing the graph: The calculator displays how the curve looks, and you can use this to understand key features of the function like intercepts and general shape.
  • Zoom tools: These features allow you to zoom in on specific parts of the graph for detailed analysis.
These steps help ensure a clearer understanding of the function, making it easier to find inverse points later.
Function Analysis
Analyzing a function is the key to understanding its properties. Each function has unique characteristics based on its equation. For the function \(f(x) = x^3 + 3x - 4\), let's look deeper.
  • Behavior: This function is a cubic polynomial. Cubic functions generally have an "S" shaped curve and continue infinitely in both vertical directions.
  • Intercepts: Functions are often analyzed by finding the x-intercepts (where \(f(x) = 0\)) and y-intercepts (where \(x = 0\)). Knowing these helps identify main features of the graph.
  • Variability: Checking how the function increases or decreases across its domain can determine where it has maximum or minimum points.
Function analysis lays the groundwork for finding inverse points which are vital in creating a complete picture of the function and its inverse.
Inverse Points
Finding inverse points involves understanding the fundamental definition of inverse functions. For a function, if \((a, b)\) lies on its graph, then \((b, a)\) will be on the graph of its inverse. This concept flips the roles of inputs and outputs.
  • Solving Equations: To locate inverse points for specific \(y\)-values (such as 0, 1, and 2 in our case), substitute these into the function and solve for \(x\). The values of \(x\) you find adjust to become \(y\) values on the inverse graph.
  • Inverse Graph Points: Thus, solving \(f(x) = 0\), \(f(x) = 1\), and \(f(x) = 2\) gives us crucial points \((0, x_1)\), \((1, x_2)\), and \((2, x_3)\) marking the inverse.
Understanding inverse points bridges the connection between a function and its inverse, highlighting the relationship in their graphs.

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Most popular questions from this chapter

For the following exercises, determine the function described and then use it to answer the question. The volume, \(V,\) of a sphere in terms of its radius,r is given by \(V(r)=\frac{4}{3} \pi r^{3}\) . Express \(r\) as a function of \(V,\) and find the radius of a sphere with volume of 200 cubic feet.

For the following exercises, use Kepler's Law, which states that the square of the time, \(T\), required for a planet to orbit the Sun varies directly with the cube of the mean distance, \(a\), that the planet is from the Sun. Using Earth's distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w\). When \(x=2\), \(z=2\), and \(w=64\), then \(y=12\). Find \(y\) when \(x=1\), \(z=3,\) and \(w=4\).

For the following exercises, list all possible rational zeros for the functions. \(f(x)=2 x^{3}+3 x^{2}-8 x+5\)

For the following exercises, determine the function described and then use it to answer the question. A container holds 100 \(\mathrm{ml}\) of a solution that is 25 \(\mathrm{ml}\) acid. If \(n \mathrm{ml}\) of a solution that is 60\(\%\) acid is added, the function \(C(n)=\frac{25+0.6 n}{100+n}\) gives the concentration, \(C,\) as a function of the number of ml added, \(n .\) Express \(n\) as a function of \(C\) and determine the number of \(m 1\) that need to be added to have a solution that is 50\(\%\) acid.
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